Math = Love: Drawing Pictures: Reflections on Problem Solving

Tuesday, April 22, 2014

Drawing Pictures: Reflections on Problem Solving

The more I teach, the more I learn.  Eventually, I'm going to become pretty great at this job.

The summer before my first year of teaching, I made a set of problem solving strategy posters to hang on the wall.  My first year of teaching, these posters had an entire bulletin board dedicated to them.  I would occasionally reference them in class, but I didn't really do anything substantial with them.  

Problem Solving Strategies Bulletin Board
This summer, I redecorated and rearranged my classroom.  In order to make room for a Star Student bulletin board, I moved the problem solving strategies to a new home underneath my SMART Board.

New Home for Problem Solving Strategy Posters
I don't even think most of my students ever even noticed that they were there.  Again, I did nothing more than post them on the wall.

If I really and truly want my students to develop their problem solving skills, I have to do more than just post the strategies.  I have to make them do the strategies.  I have to make them practice the strategies.  I have to remind them of the strategies.  But, it's more than that.  I have to give them challenging problems that will force them to utilize the strategies.  If I'm not challenging them, if I'm not holding to them to a high standard, then it's pointless to even post the strategies.

This year, I opted to skip the unit on ratios and proportions in Algebra 1.  Under Common Core, ratios and proportions will be a middle school unit.  And, I knew that my students had seen these types of problems in middle school.  Since this is the last year of testing our old standards in Oklahoma, students were asked several problems involving ratios, proportions, and percents on their end-of-instruction exam.  During our few weeks of EOI test review, I threw these ratio/proportion problems up on the board to see how my students would handle them.

Here's one of the problems I chose from questions released by the Oklahoma Department of Education:

At a candle store, the ratio of green candles to red candles is 2 to 5.  The store has 4,900 candles.  How many candles are red?

This is a tricky question.  When students see that 2 and 5, their gut instinct is to make a ratio out of them and then form a proportion.  4,900 isn't the number of green or red candles, though.  It's the total number of candles.  So, we need a ratio that deals with the total number of candles.  We must represent the ratio of red candles to total candles as 5/(2+5).

Knowing that students were likely to be tripped up by this problem, I urged them to draw a picture before performing any calculations.  It was a simple request.  It was a request that I should make more often.

Draw me a picture.

A.K.A. Use a problem solving strategy!

And, don't just use it.  Use it, and then show the class how you used it.

Oh, you used a strategy but ended up doing something incorrectly?  Awesome!  You just provided me with insight to your thought process, and it's an excellent learning experience for the class.

I didn't realize just how powerful that question would be until students started holding up their dry erase boards with their pictures.

Let's just say that the way I approached this problem and the way many of my students approached this problem was differently.  When I asked for a picture, I had a certain picture already drawn in my mind.  So, when students started holding up pictures that were unlike what I expected, it was an awesome experience.  I had to grab my camera and document this experience!

I wasn't expecting a circle graph.  But, it works!  And, I almost wish I had thought of drawing my picture like this.

Circle Graph

Another student thought in terms of writing the ratio using a colon.  Many students in the class could relate to this.  Again, it wasn't what I was going for.  But, that doesn't make it a viable picture.


This student also used the colon format to write their ratios.  She has made the common error that I predicted upon picking out this problem.


Of course, some students drew pictures that were creative but not exactly mathematical.

I present to you: Sherrie's Candle Store

Picture of Candle Store

When I drew my picture on the board, I drew two green candles and five red candles inside a lovely store.  (You can tell me just how awesome of an artist I am in the comments!)  Realizing that red and green candles were the color of Christmas candles, I changed the name of my store to the "Christmas Candle Store."

Next, I drew the sideways squiggly bracket, and I asked students how many candles were represented by the picture I drew.  4900 candles.  How many candles did I draw?  7 candles.  So, how many actual candles must each drawn candle represent?  700 candles!  Once this discovery was made, my students were quick to point out that the correct answer was (d) 3,500 red candles.  Why?  After a satisfactory answer was given, I drew the circle graph that one of my students had drawn on their board on the SMART Board.

What if your picture looked like this?  What would you do next?  We walked through the same solution process using a different picture.

Illustrating the Question

Why have I never done this before?  I draw pictures on the board all the time.  But, I don't have my students do the same.  I've been cheating them out of a learning experience by drawing the picture for them.  I've been cheating myself out of a learning experience by not letting them draw their own pictures.  I guess I've always been afraid that their pictures would be wrong.  I've been afraid that they would make mistakes.

This stops now.  Will they make mistakes?  Certainly.  Mistakes are how we learn.  I've always thought that, but my actions haven't been reflecting that in my classroom.

I'm currently reading a book on how to organize my house/life by setting up daily/weekly/monthly routines.  Maybe the best thing to come out of this book was a quote from the author's husband's geometry teacher: "Anything worth doing is worth doing wrong."

Do I really believe that?  Do I teach like that is true?  Because it definitely is.  And, I need to make this my mantra.  If a problem is truly worth doing, then it is worth it for my students to do it wrong.  There is something to be gleaned in the process of analyzing errors.

To try next year:

Problem Solving Strategy Gallery Walk

1.  Pick an awesome, thought-provoking problem for which the answer is not immediately apparent.
2.  List various (applicable) problem-solving strategies on strips of paper.
3.  Let each student draw a strip of paper.  Give them 3/5/10 minutes to apply that strategy to the problem.
4.  On a sheet of paper, they must illustrate how the strategy can be used.  However, they CANNOT solve the problem.  They should draw the picture, make the table, write the equation.  They should not answer the question.  That will come later.
5.  Hang the papers around the classroom.
6.  Give each student 3/4/5 post-it notes.
7.  Students must walk around the room and leave constructive feedback on the strategies employed by their classmates.
8.  Once feedback has been given, group students by strategy.  All of the students who drew a picture sit together.  All of the students who made a table sit together.  All of the students who wrote an equation sit together.
9.  Each group takes the feedback provided by their classmates and works the problem together, coming to a singular solution.  (Students may work problems individually first and then come together for a group solution.)
10.  Each group presents their solution and explains how they used their problem solving strategy.

Why I think I will like this:
* Emphasis on multiple avenues available to solve a problem
* Students giving students feedback - there's not enough of this in my classroom
* It will force me to ask more deeper, more complex, more thought-provoking questions

Thoughts?  Feedback?

I realize that problems are going to take longer this way.  We might spend an entire 50-minute period on one question.  But, is that a bad thing?  I've spent the last two years rushing through class period after class period to make sure I "cover" everything.  No wonder they forget everything from the first semester by the time the end of year test comes around.  Students learn and internalize what we dwell on.  And, problem solving strategies is something I need to dwell on.  Few of my students will ever factor polynomials or graph exponentials as part of their daily lives.  But, they will all face problems that need to be solved, and they will need tools to help them sort through the many options available to them.  If they leave my classroom as critical thinkers and problem solvers who persevere until a solution is found, I have done my job.

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